from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,38]))
pari: [g,chi] = znchar(Mod(121,245))
Basic properties
Modulus: | \(245\) | |
Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{49}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 245.q
\(\chi_{245}(11,\cdot)\) \(\chi_{245}(16,\cdot)\) \(\chi_{245}(46,\cdot)\) \(\chi_{245}(51,\cdot)\) \(\chi_{245}(81,\cdot)\) \(\chi_{245}(86,\cdot)\) \(\chi_{245}(121,\cdot)\) \(\chi_{245}(151,\cdot)\) \(\chi_{245}(156,\cdot)\) \(\chi_{245}(186,\cdot)\) \(\chi_{245}(191,\cdot)\) \(\chi_{245}(221,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((197,101)\) → \((1,e\left(\frac{19}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
\( \chi_{ 245 }(121, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)